3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
42 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
45 // default ctor, dtor, copy ctor, assignment operator and helpers:
48 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
49 matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
51 debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
55 void matrix::copy(const matrix & other)
57 inherited::copy(other);
60 m = other.m; // STL's vector copying invoked here
63 DEFAULT_DESTROY(matrix)
71 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
73 * @param r number of rows
74 * @param c number of cols */
75 matrix::matrix(unsigned r, unsigned c)
76 : inherited(TINFO_matrix), row(r), col(c)
78 debugmsg("matrix ctor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
79 m.resize(r*c, _ex0());
84 /** Ctor from representation, for internal use only. */
85 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
86 : inherited(TINFO_matrix), row(r), col(c), m(m2)
88 debugmsg("matrix ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
91 /** Construct matrix from (flat) list of elements. If the list has fewer
92 * elements than the matrix, the remaining matrix elements are set to zero.
93 * If the list has more elements than the matrix, the excessive elements are
95 matrix::matrix(unsigned r, unsigned c, const lst & l)
96 : inherited(TINFO_matrix), row(r), col(c)
98 debugmsg("matrix ctor from unsigned,unsigned,lst",LOGLEVEL_CONSTRUCT);
99 m.resize(r*c, _ex0());
101 for (unsigned i=0; i<l.nops(); i++) {
105 break; // matrix smaller than list: throw away excessive elements
114 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
116 debugmsg("matrix ctor from archive_node", LOGLEVEL_CONSTRUCT);
117 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
118 throw (std::runtime_error("unknown matrix dimensions in archive"));
119 m.reserve(row * col);
120 for (unsigned int i=0; true; i++) {
122 if (n.find_ex("m", e, sym_lst, i))
129 void matrix::archive(archive_node &n) const
131 inherited::archive(n);
132 n.add_unsigned("row", row);
133 n.add_unsigned("col", col);
134 exvector::const_iterator i = m.begin(), iend = m.end();
141 DEFAULT_UNARCHIVE(matrix)
144 // functions overriding virtual functions from bases classes
149 void matrix::print(const print_context & c, unsigned level) const
151 debugmsg("matrix print", LOGLEVEL_PRINT);
153 if (is_of_type(c, print_tree)) {
155 inherited::print(c, level);
160 for (unsigned y=0; y<row-1; ++y) {
162 for (unsigned x=0; x<col-1; ++x) {
166 m[col*(y+1)-1].print(c);
170 for (unsigned x=0; x<col-1; ++x) {
171 m[(row-1)*col+x].print(c);
174 m[row*col-1].print(c);
180 /** nops is defined to be rows x columns. */
181 unsigned matrix::nops() const
186 /** returns matrix entry at position (i/col, i%col). */
187 ex matrix::op(int i) const
192 /** returns matrix entry at position (i/col, i%col). */
193 ex & matrix::let_op(int i)
196 GINAC_ASSERT(i<nops());
201 /** expands the elements of a matrix entry by entry. */
202 ex matrix::expand(unsigned options) const
204 exvector tmp(row*col);
205 for (unsigned i=0; i<row*col; ++i)
206 tmp[i] = m[i].expand(options);
208 return matrix(row, col, tmp);
211 /** Evaluate matrix entry by entry. */
212 ex matrix::eval(int level) const
214 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
216 // check if we have to do anything at all
217 if ((level==1)&&(flags & status_flags::evaluated))
221 if (level == -max_recursion_level)
222 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
224 // eval() entry by entry
225 exvector m2(row*col);
227 for (unsigned r=0; r<row; ++r)
228 for (unsigned c=0; c<col; ++c)
229 m2[r*col+c] = m[r*col+c].eval(level);
231 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
232 status_flags::evaluated );
235 /** Evaluate matrix numerically entry by entry. */
236 ex matrix::evalf(int level) const
238 debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
240 // check if we have to do anything at all
245 if (level == -max_recursion_level) {
246 throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
249 // evalf() entry by entry
250 exvector m2(row*col);
252 for (unsigned r=0; r<row; ++r)
253 for (unsigned c=0; c<col; ++c)
254 m2[r*col+c] = m[r*col+c].evalf(level);
256 return matrix(row, col, m2);
259 ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
261 exvector m2(row * col);
262 for (unsigned r=0; r<row; ++r)
263 for (unsigned c=0; c<col; ++c)
264 m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
266 return ex(matrix(row, col, m2)).bp->basic::subs(ls, lr, no_pattern);
271 int matrix::compare_same_type(const basic & other) const
273 GINAC_ASSERT(is_exactly_of_type(other, matrix));
274 const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
276 // compare number of rows
278 return row < o.rows() ? -1 : 1;
280 // compare number of columns
282 return col < o.cols() ? -1 : 1;
284 // equal number of rows and columns, compare individual elements
286 for (unsigned r=0; r<row; ++r) {
287 for (unsigned c=0; c<col; ++c) {
288 cmpval = ((*this)(r,c)).compare(o(r,c));
289 if (cmpval!=0) return cmpval;
292 // all elements are equal => matrices are equal;
296 /** Automatic symbolic evaluation of an indexed matrix. */
297 ex matrix::eval_indexed(const basic & i) const
299 GINAC_ASSERT(is_of_type(i, indexed));
300 GINAC_ASSERT(is_ex_of_type(i.op(0), matrix));
302 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
307 // One index, must be one-dimensional vector
308 if (row != 1 && col != 1)
309 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
311 const idx & i1 = ex_to_idx(i.op(1));
316 if (!i1.get_dim().is_equal(row))
317 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
319 // Index numeric -> return vector element
320 if (all_indices_unsigned) {
321 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
323 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
324 return (*this)(n1, 0);
330 if (!i1.get_dim().is_equal(col))
331 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
333 // Index numeric -> return vector element
334 if (all_indices_unsigned) {
335 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
337 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
338 return (*this)(0, n1);
342 } else if (i.nops() == 3) {
345 const idx & i1 = ex_to_idx(i.op(1));
346 const idx & i2 = ex_to_idx(i.op(2));
348 if (!i1.get_dim().is_equal(row))
349 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
350 if (!i2.get_dim().is_equal(col))
351 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
353 // Pair of dummy indices -> compute trace
354 if (is_dummy_pair(i1, i2))
357 // Both indices numeric -> return matrix element
358 if (all_indices_unsigned) {
359 unsigned n1 = ex_to_numeric(i1.get_value()).to_int(), n2 = ex_to_numeric(i2.get_value()).to_int();
361 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
363 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
364 return (*this)(n1, n2);
368 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
373 /** Sum of two indexed matrices. */
374 ex matrix::add_indexed(const ex & self, const ex & other) const
376 GINAC_ASSERT(is_ex_of_type(self, indexed));
377 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
378 GINAC_ASSERT(is_ex_of_type(other, indexed));
379 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
381 // Only add two matrices
382 if (is_ex_of_type(other.op(0), matrix)) {
383 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
385 const matrix &self_matrix = ex_to_matrix(self.op(0));
386 const matrix &other_matrix = ex_to_matrix(other.op(0));
388 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
390 if (self_matrix.row == other_matrix.row)
391 return indexed(self_matrix.add(other_matrix), self.op(1));
392 else if (self_matrix.row == other_matrix.col)
393 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
395 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
397 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
398 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
399 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
400 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
405 // Don't know what to do, return unevaluated sum
409 /** Product of an indexed matrix with a number. */
410 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
412 GINAC_ASSERT(is_ex_of_type(self, indexed));
413 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
414 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
416 const matrix &self_matrix = ex_to_matrix(self.op(0));
418 if (self.nops() == 2)
419 return indexed(self_matrix.mul(other), self.op(1));
420 else // self.nops() == 3
421 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
424 /** Contraction of an indexed matrix with something else. */
425 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
427 GINAC_ASSERT(is_ex_of_type(*self, indexed));
428 GINAC_ASSERT(is_ex_of_type(*other, indexed));
429 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
430 GINAC_ASSERT(is_ex_of_type(self->op(0), matrix));
432 // Only contract with other matrices
433 if (!is_ex_of_type(other->op(0), matrix))
436 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
438 const matrix &self_matrix = ex_to_matrix(self->op(0));
439 const matrix &other_matrix = ex_to_matrix(other->op(0));
441 if (self->nops() == 2) {
442 unsigned self_dim = (self_matrix.col == 1) ? self_matrix.row : self_matrix.col;
444 if (other->nops() == 2) { // vector * vector (scalar product)
445 unsigned other_dim = (other_matrix.col == 1) ? other_matrix.row : other_matrix.col;
447 if (self_matrix.col == 1) {
448 if (other_matrix.col == 1) {
449 // Column vector * column vector, transpose first vector
450 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
452 // Column vector * row vector, swap factors
453 *self = other_matrix.mul(self_matrix)(0, 0);
456 if (other_matrix.col == 1) {
457 // Row vector * column vector, perfect
458 *self = self_matrix.mul(other_matrix)(0, 0);
460 // Row vector * row vector, transpose second vector
461 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
467 } else { // vector * matrix
469 // B_i * A_ij = (B*A)_j (B is row vector)
470 if (is_dummy_pair(self->op(1), other->op(1))) {
471 if (self_matrix.row == 1)
472 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
474 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
479 // B_j * A_ij = (A*B)_i (B is column vector)
480 if (is_dummy_pair(self->op(1), other->op(2))) {
481 if (self_matrix.col == 1)
482 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
484 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
490 } else if (other->nops() == 3) { // matrix * matrix
492 // A_ij * B_jk = (A*B)_ik
493 if (is_dummy_pair(self->op(2), other->op(1))) {
494 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
499 // A_ij * B_kj = (A*Btrans)_ik
500 if (is_dummy_pair(self->op(2), other->op(2))) {
501 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
506 // A_ji * B_jk = (Atrans*B)_ik
507 if (is_dummy_pair(self->op(1), other->op(1))) {
508 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
513 // A_ji * B_kj = (B*A)_ki
514 if (is_dummy_pair(self->op(1), other->op(2))) {
515 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
526 // non-virtual functions in this class
533 * @exception logic_error (incompatible matrices) */
534 matrix matrix::add(const matrix & other) const
536 if (col != other.col || row != other.row)
537 throw (std::logic_error("matrix::add(): incompatible matrices"));
539 exvector sum(this->m);
540 exvector::iterator i;
541 exvector::const_iterator ci;
542 for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
545 return matrix(row,col,sum);
549 /** Difference of matrices.
551 * @exception logic_error (incompatible matrices) */
552 matrix matrix::sub(const matrix & other) const
554 if (col != other.col || row != other.row)
555 throw (std::logic_error("matrix::sub(): incompatible matrices"));
557 exvector dif(this->m);
558 exvector::iterator i;
559 exvector::const_iterator ci;
560 for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
563 return matrix(row,col,dif);
567 /** Product of matrices.
569 * @exception logic_error (incompatible matrices) */
570 matrix matrix::mul(const matrix & other) const
572 if (this->cols() != other.rows())
573 throw (std::logic_error("matrix::mul(): incompatible matrices"));
575 exvector prod(this->rows()*other.cols());
577 for (unsigned r1=0; r1<this->rows(); ++r1) {
578 for (unsigned c=0; c<this->cols(); ++c) {
579 if (m[r1*col+c].is_zero())
581 for (unsigned r2=0; r2<other.cols(); ++r2)
582 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
585 return matrix(row, other.col, prod);
589 /** Product of matrix and scalar. */
590 matrix matrix::mul(const numeric & other) const
592 exvector prod(row * col);
594 for (unsigned r=0; r<row; ++r)
595 for (unsigned c=0; c<col; ++c)
596 prod[r*col+c] = m[r*col+c] * other;
598 return matrix(row, col, prod);
602 /** operator() to access elements.
604 * @param ro row of element
605 * @param co column of element
606 * @exception range_error (index out of range) */
607 const ex & matrix::operator() (unsigned ro, unsigned co) const
609 if (ro>=row || co>=col)
610 throw (std::range_error("matrix::operator(): index out of range"));
616 /** Set individual elements manually.
618 * @exception range_error (index out of range) */
619 matrix & matrix::set(unsigned ro, unsigned co, ex value)
621 if (ro>=row || co>=col)
622 throw (std::range_error("matrix::set(): index out of range"));
624 ensure_if_modifiable();
625 m[ro*col+co] = value;
630 /** Transposed of an m x n matrix, producing a new n x m matrix object that
631 * represents the transposed. */
632 matrix matrix::transpose(void) const
634 exvector trans(this->cols()*this->rows());
636 for (unsigned r=0; r<this->cols(); ++r)
637 for (unsigned c=0; c<this->rows(); ++c)
638 trans[r*this->rows()+c] = m[c*this->cols()+r];
640 return matrix(this->cols(),this->rows(),trans);
644 /** Determinant of square matrix. This routine doesn't actually calculate the
645 * determinant, it only implements some heuristics about which algorithm to
646 * run. If all the elements of the matrix are elements of an integral domain
647 * the determinant is also in that integral domain and the result is expanded
648 * only. If one or more elements are from a quotient field the determinant is
649 * usually also in that quotient field and the result is normalized before it
650 * is returned. This implies that the determinant of the symbolic 2x2 matrix
651 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
652 * behaves like MapleV and unlike Mathematica.)
654 * @param algo allows to chose an algorithm
655 * @return the determinant as a new expression
656 * @exception logic_error (matrix not square)
657 * @see determinant_algo */
658 ex matrix::determinant(unsigned algo) const
661 throw (std::logic_error("matrix::determinant(): matrix not square"));
662 GINAC_ASSERT(row*col==m.capacity());
664 // Gather some statistical information about this matrix:
665 bool numeric_flag = true;
666 bool normal_flag = false;
667 unsigned sparse_count = 0; // counts non-zero elements
668 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
669 lst srl; // symbol replacement list
670 ex rtest = (*r).to_rational(srl);
671 if (!rtest.is_zero())
673 if (!rtest.info(info_flags::numeric))
674 numeric_flag = false;
675 if (!rtest.info(info_flags::crational_polynomial) &&
676 rtest.info(info_flags::rational_function))
680 // Here is the heuristics in case this routine has to decide:
681 if (algo == determinant_algo::automatic) {
682 // Minor expansion is generally a good guess:
683 algo = determinant_algo::laplace;
684 // Does anybody know when a matrix is really sparse?
685 // Maybe <~row/2.236 nonzero elements average in a row?
686 if (row>3 && 5*sparse_count<=row*col)
687 algo = determinant_algo::bareiss;
688 // Purely numeric matrix can be handled by Gauss elimination.
689 // This overrides any prior decisions.
691 algo = determinant_algo::gauss;
694 // Trap the trivial case here, since some algorithms don't like it
696 // for consistency with non-trivial determinants...
698 return m[0].normal();
700 return m[0].expand();
703 // Compute the determinant
705 case determinant_algo::gauss: {
708 int sign = tmp.gauss_elimination(true);
709 for (unsigned d=0; d<row; ++d)
710 det *= tmp.m[d*col+d];
712 return (sign*det).normal();
714 return (sign*det).normal().expand();
716 case determinant_algo::bareiss: {
719 sign = tmp.fraction_free_elimination(true);
721 return (sign*tmp.m[row*col-1]).normal();
723 return (sign*tmp.m[row*col-1]).expand();
725 case determinant_algo::divfree: {
728 sign = tmp.division_free_elimination(true);
731 ex det = tmp.m[row*col-1];
732 // factor out accumulated bogus slag
733 for (unsigned d=0; d<row-2; ++d)
734 for (unsigned j=0; j<row-d-2; ++j)
735 det = (det/tmp.m[d*col+d]).normal();
738 case determinant_algo::laplace:
740 // This is the minor expansion scheme. We always develop such
741 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
742 // rightmost column. For this to be efficient it turns out that
743 // the emptiest columns (i.e. the ones with most zeros) should be
744 // the ones on the right hand side. Therefore we presort the
745 // columns of the matrix:
746 typedef std::pair<unsigned,unsigned> uintpair;
747 std::vector<uintpair> c_zeros; // number of zeros in column
748 for (unsigned c=0; c<col; ++c) {
750 for (unsigned r=0; r<row; ++r)
751 if (m[r*col+c].is_zero())
753 c_zeros.push_back(uintpair(acc,c));
755 sort(c_zeros.begin(),c_zeros.end());
756 std::vector<unsigned> pre_sort;
757 for (std::vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
758 pre_sort.push_back(i->second);
759 int sign = permutation_sign(pre_sort);
760 exvector result(row*col); // represents sorted matrix
762 for (std::vector<unsigned>::iterator i=pre_sort.begin();
765 for (unsigned r=0; r<row; ++r)
766 result[r*col+c] = m[r*col+(*i)];
770 return (sign*matrix(row,col,result).determinant_minor()).normal();
772 return sign*matrix(row,col,result).determinant_minor();
778 /** Trace of a matrix. The result is normalized if it is in some quotient
779 * field and expanded only otherwise. This implies that the trace of the
780 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
782 * @return the sum of diagonal elements
783 * @exception logic_error (matrix not square) */
784 ex matrix::trace(void) const
787 throw (std::logic_error("matrix::trace(): matrix not square"));
790 for (unsigned r=0; r<col; ++r)
793 if (tr.info(info_flags::rational_function) &&
794 !tr.info(info_flags::crational_polynomial))
801 /** Characteristic Polynomial. Following mathematica notation the
802 * characteristic polynomial of a matrix M is defined as the determiant of
803 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
804 * as M. Note that some CASs define it with a sign inside the determinant
805 * which gives rise to an overall sign if the dimension is odd. This method
806 * returns the characteristic polynomial collected in powers of lambda as a
809 * @return characteristic polynomial as new expression
810 * @exception logic_error (matrix not square)
811 * @see matrix::determinant() */
812 ex matrix::charpoly(const symbol & lambda) const
815 throw (std::logic_error("matrix::charpoly(): matrix not square"));
817 bool numeric_flag = true;
818 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
819 if (!(*r).info(info_flags::numeric)) {
820 numeric_flag = false;
824 // The pure numeric case is traditionally rather common. Hence, it is
825 // trapped and we use Leverrier's algorithm which goes as row^3 for
826 // every coefficient. The expensive part is the matrix multiplication.
830 ex poly = power(lambda,row)-c*power(lambda,row-1);
831 for (unsigned i=1; i<row; ++i) {
832 for (unsigned j=0; j<row; ++j)
835 c = B.trace()/ex(i+1);
836 poly -= c*power(lambda,row-i-1);
845 for (unsigned r=0; r<col; ++r)
846 M.m[r*col+r] -= lambda;
848 return M.determinant().collect(lambda);
852 /** Inverse of this matrix.
854 * @return the inverted matrix
855 * @exception logic_error (matrix not square)
856 * @exception runtime_error (singular matrix) */
857 matrix matrix::inverse(void) const
860 throw (std::logic_error("matrix::inverse(): matrix not square"));
862 // This routine actually doesn't do anything fancy at all. We compute the
863 // inverse of the matrix A by solving the system A * A^{-1} == Id.
865 // First populate the identity matrix supposed to become the right hand side.
866 matrix identity(row,col);
867 for (unsigned i=0; i<row; ++i)
868 identity.set(i,i,_ex1());
870 // Populate a dummy matrix of variables, just because of compatibility with
871 // matrix::solve() which wants this (for compatibility with under-determined
872 // systems of equations).
873 matrix vars(row,col);
874 for (unsigned r=0; r<row; ++r)
875 for (unsigned c=0; c<col; ++c)
876 vars.set(r,c,symbol());
880 sol = this->solve(vars,identity);
881 } catch (const std::runtime_error & e) {
882 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
883 throw (std::runtime_error("matrix::inverse(): singular matrix"));
891 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
892 * side by applying an elimination scheme to the augmented matrix.
894 * @param vars n x p matrix, all elements must be symbols
895 * @param rhs m x p matrix
896 * @return n x p solution matrix
897 * @exception logic_error (incompatible matrices)
898 * @exception invalid_argument (1st argument must be matrix of symbols)
899 * @exception runtime_error (inconsistent linear system)
901 matrix matrix::solve(const matrix & vars,
905 const unsigned m = this->rows();
906 const unsigned n = this->cols();
907 const unsigned p = rhs.cols();
910 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
911 throw (std::logic_error("matrix::solve(): incompatible matrices"));
912 for (unsigned ro=0; ro<n; ++ro)
913 for (unsigned co=0; co<p; ++co)
914 if (!vars(ro,co).info(info_flags::symbol))
915 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
917 // build the augmented matrix of *this with rhs attached to the right
919 for (unsigned r=0; r<m; ++r) {
920 for (unsigned c=0; c<n; ++c)
921 aug.m[r*(n+p)+c] = this->m[r*n+c];
922 for (unsigned c=0; c<p; ++c)
923 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
926 // Gather some statistical information about the augmented matrix:
927 bool numeric_flag = true;
928 for (exvector::const_iterator r=aug.m.begin(); r!=aug.m.end(); ++r) {
929 if (!(*r).info(info_flags::numeric))
930 numeric_flag = false;
933 // Here is the heuristics in case this routine has to decide:
934 if (algo == solve_algo::automatic) {
935 // Bareiss (fraction-free) elimination is generally a good guess:
936 algo = solve_algo::bareiss;
937 // For m<3, Bareiss elimination is equivalent to division free
938 // elimination but has more logistic overhead
940 algo = solve_algo::divfree;
941 // This overrides any prior decisions.
943 algo = solve_algo::gauss;
946 // Eliminate the augmented matrix:
948 case solve_algo::gauss:
949 aug.gauss_elimination();
951 case solve_algo::divfree:
952 aug.division_free_elimination();
954 case solve_algo::bareiss:
956 aug.fraction_free_elimination();
959 // assemble the solution matrix:
961 for (unsigned co=0; co<p; ++co) {
962 unsigned last_assigned_sol = n+1;
963 for (int r=m-1; r>=0; --r) {
964 unsigned fnz = 1; // first non-zero in row
965 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
968 // row consists only of zeros, corresponding rhs must be 0, too
969 if (!aug.m[r*(n+p)+n+co].is_zero()) {
970 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
973 // assign solutions for vars between fnz+1 and
974 // last_assigned_sol-1: free parameters
975 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
976 sol.set(c,co,vars.m[c*p+co]);
977 ex e = aug.m[r*(n+p)+n+co];
978 for (unsigned c=fnz; c<n; ++c)
979 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
981 (e/(aug.m[r*(n+p)+(fnz-1)])).normal());
982 last_assigned_sol = fnz;
985 // assign solutions for vars between 1 and
986 // last_assigned_sol-1: free parameters
987 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
988 sol.set(ro,co,vars(ro,co));
997 /** Recursive determinant for small matrices having at least one symbolic
998 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
999 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1000 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1001 * is better than elimination schemes for matrices of sparse multivariate
1002 * polynomials and also for matrices of dense univariate polynomials if the
1003 * matrix' dimesion is larger than 7.
1005 * @return the determinant as a new expression (in expanded form)
1006 * @see matrix::determinant() */
1007 ex matrix::determinant_minor(void) const
1009 // for small matrices the algorithm does not make any sense:
1010 const unsigned n = this->cols();
1012 return m[0].expand();
1014 return (m[0]*m[3]-m[2]*m[1]).expand();
1016 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1017 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1018 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1020 // This algorithm can best be understood by looking at a naive
1021 // implementation of Laplace-expansion, like this one:
1023 // matrix minorM(this->rows()-1,this->cols()-1);
1024 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1025 // // shortcut if element(r1,0) vanishes
1026 // if (m[r1*col].is_zero())
1028 // // assemble the minor matrix
1029 // for (unsigned r=0; r<minorM.rows(); ++r) {
1030 // for (unsigned c=0; c<minorM.cols(); ++c) {
1032 // minorM.set(r,c,m[r*col+c+1]);
1034 // minorM.set(r,c,m[(r+1)*col+c+1]);
1037 // // recurse down and care for sign:
1039 // det -= m[r1*col] * minorM.determinant_minor();
1041 // det += m[r1*col] * minorM.determinant_minor();
1043 // return det.expand();
1044 // What happens is that while proceeding down many of the minors are
1045 // computed more than once. In particular, there are binomial(n,k)
1046 // kxk minors and each one is computed factorial(n-k) times. Therefore
1047 // it is reasonable to store the results of the minors. We proceed from
1048 // right to left. At each column c we only need to retrieve the minors
1049 // calculated in step c-1. We therefore only have to store at most
1050 // 2*binomial(n,n/2) minors.
1052 // Unique flipper counter for partitioning into minors
1053 std::vector<unsigned> Pkey;
1055 // key for minor determinant (a subpartition of Pkey)
1056 std::vector<unsigned> Mkey;
1058 // we store our subminors in maps, keys being the rows they arise from
1059 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1060 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1064 // initialize A with last column:
1065 for (unsigned r=0; r<n; ++r) {
1066 Pkey.erase(Pkey.begin(),Pkey.end());
1068 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1070 // proceed from right to left through matrix
1071 for (int c=n-2; c>=0; --c) {
1072 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1073 Mkey.erase(Mkey.begin(),Mkey.end());
1074 for (unsigned i=0; i<n-c; ++i)
1076 unsigned fc = 0; // controls logic for our strange flipper counter
1079 for (unsigned r=0; r<n-c; ++r) {
1080 // maybe there is nothing to do?
1081 if (m[Pkey[r]*n+c].is_zero())
1083 // create the sorted key for all possible minors
1084 Mkey.erase(Mkey.begin(),Mkey.end());
1085 for (unsigned i=0; i<n-c; ++i)
1087 Mkey.push_back(Pkey[i]);
1088 // Fetch the minors and compute the new determinant
1090 det -= m[Pkey[r]*n+c]*A[Mkey];
1092 det += m[Pkey[r]*n+c]*A[Mkey];
1094 // prevent build-up of deep nesting of expressions saves time:
1096 // store the new determinant at its place in B:
1098 B.insert(Rmap_value(Pkey,det));
1099 // increment our strange flipper counter
1100 for (fc=n-c; fc>0; --fc) {
1102 if (Pkey[fc-1]<fc+c)
1106 for (unsigned j=fc; j<n-c; ++j)
1107 Pkey[j] = Pkey[j-1]+1;
1109 // next column, so change the role of A and B:
1118 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1119 * matrix into an upper echelon form. The algorithm is ok for matrices
1120 * with numeric coefficients but quite unsuited for symbolic matrices.
1122 * @param det may be set to true to save a lot of space if one is only
1123 * interested in the diagonal elements (i.e. for calculating determinants).
1124 * The others are set to zero in this case.
1125 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1126 * number of rows was swapped and 0 if the matrix is singular. */
1127 int matrix::gauss_elimination(const bool det)
1129 ensure_if_modifiable();
1130 const unsigned m = this->rows();
1131 const unsigned n = this->cols();
1132 GINAC_ASSERT(!det || n==m);
1136 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1137 int indx = pivot(r0, r1, true);
1141 return 0; // leaves *this in a messy state
1146 for (unsigned r2=r0+1; r2<m; ++r2) {
1147 if (!this->m[r2*n+r1].is_zero()) {
1148 // yes, there is something to do in this row
1149 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1150 for (unsigned c=r1+1; c<n; ++c) {
1151 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1152 if (!this->m[r2*n+c].info(info_flags::numeric))
1153 this->m[r2*n+c] = this->m[r2*n+c].normal();
1156 // fill up left hand side with zeros
1157 for (unsigned c=0; c<=r1; ++c)
1158 this->m[r2*n+c] = _ex0();
1161 // save space by deleting no longer needed elements
1162 for (unsigned c=r0+1; c<n; ++c)
1163 this->m[r0*n+c] = _ex0();
1173 /** Perform the steps of division free elimination to bring the m x n matrix
1174 * into an upper echelon form.
1176 * @param det may be set to true to save a lot of space if one is only
1177 * interested in the diagonal elements (i.e. for calculating determinants).
1178 * The others are set to zero in this case.
1179 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1180 * number of rows was swapped and 0 if the matrix is singular. */
1181 int matrix::division_free_elimination(const bool det)
1183 ensure_if_modifiable();
1184 const unsigned m = this->rows();
1185 const unsigned n = this->cols();
1186 GINAC_ASSERT(!det || n==m);
1190 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1191 int indx = pivot(r0, r1, true);
1195 return 0; // leaves *this in a messy state
1200 for (unsigned r2=r0+1; r2<m; ++r2) {
1201 for (unsigned c=r1+1; c<n; ++c)
1202 this->m[r2*n+c] = (this->m[r0*n+r1]*this->m[r2*n+c] - this->m[r2*n+r1]*this->m[r0*n+c]).expand();
1203 // fill up left hand side with zeros
1204 for (unsigned c=0; c<=r1; ++c)
1205 this->m[r2*n+c] = _ex0();
1208 // save space by deleting no longer needed elements
1209 for (unsigned c=r0+1; c<n; ++c)
1210 this->m[r0*n+c] = _ex0();
1220 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1221 * the matrix into an upper echelon form. Fraction free elimination means
1222 * that divide is used straightforwardly, without computing GCDs first. This
1223 * is possible, since we know the divisor at each step.
1225 * @param det may be set to true to save a lot of space if one is only
1226 * interested in the last element (i.e. for calculating determinants). The
1227 * others are set to zero in this case.
1228 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1229 * number of rows was swapped and 0 if the matrix is singular. */
1230 int matrix::fraction_free_elimination(const bool det)
1233 // (single-step fraction free elimination scheme, already known to Jordan)
1235 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1236 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1238 // Bareiss (fraction-free) elimination in addition divides that element
1239 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1240 // Sylvester determinant that this really divides m[k+1](r,c).
1242 // We also allow rational functions where the original prove still holds.
1243 // However, we must care for numerator and denominator separately and
1244 // "manually" work in the integral domains because of subtle cancellations
1245 // (see below). This blows up the bookkeeping a bit and the formula has
1246 // to be modified to expand like this (N{x} stands for numerator of x,
1247 // D{x} for denominator of x):
1248 // N{m[k+1](r,c)} = N{m[k](k,k)}*N{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
1249 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1250 // D{m[k+1](r,c)} = D{m[k](k,k)}*D{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
1251 // where for k>1 we now divide N{m[k+1](r,c)} by
1252 // N{m[k-1](k-1,k-1)}
1253 // and D{m[k+1](r,c)} by
1254 // D{m[k-1](k-1,k-1)}.
1256 ensure_if_modifiable();
1257 const unsigned m = this->rows();
1258 const unsigned n = this->cols();
1259 GINAC_ASSERT(!det || n==m);
1268 // We populate temporary matrices to subsequently operate on. There is
1269 // one holding numerators and another holding denominators of entries.
1270 // This is a must since the evaluator (or even earlier mul's constructor)
1271 // might cancel some trivial element which causes divide() to fail. The
1272 // elements are normalized first (yes, even though this algorithm doesn't
1273 // need GCDs) since the elements of *this might be unnormalized, which
1274 // makes things more complicated than they need to be.
1275 matrix tmp_n(*this);
1276 matrix tmp_d(m,n); // for denominators, if needed
1277 lst srl; // symbol replacement list
1278 exvector::iterator it = this->m.begin();
1279 exvector::iterator tmp_n_it = tmp_n.m.begin();
1280 exvector::iterator tmp_d_it = tmp_d.m.begin();
1281 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
1282 (*tmp_n_it) = (*it).normal().to_rational(srl);
1283 (*tmp_d_it) = (*tmp_n_it).denom();
1284 (*tmp_n_it) = (*tmp_n_it).numer();
1288 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1289 int indx = tmp_n.pivot(r0, r1, true);
1298 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1299 for (unsigned c=r1; c<n; ++c)
1300 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1302 for (unsigned r2=r0+1; r2<m; ++r2) {
1303 for (unsigned c=r1+1; c<n; ++c) {
1304 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1305 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1306 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1307 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1308 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1309 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1310 bool check = divide(dividend_n, divisor_n,
1311 tmp_n.m[r2*n+c], true);
1312 check &= divide(dividend_d, divisor_d,
1313 tmp_d.m[r2*n+c], true);
1314 GINAC_ASSERT(check);
1316 // fill up left hand side with zeros
1317 for (unsigned c=0; c<=r1; ++c)
1318 tmp_n.m[r2*n+c] = _ex0();
1320 if ((r1<n-1)&&(r0<m-1)) {
1321 // compute next iteration's divisor
1322 divisor_n = tmp_n.m[r0*n+r1].expand();
1323 divisor_d = tmp_d.m[r0*n+r1].expand();
1325 // save space by deleting no longer needed elements
1326 for (unsigned c=0; c<n; ++c) {
1327 tmp_n.m[r0*n+c] = _ex0();
1328 tmp_d.m[r0*n+c] = _ex1();
1335 // repopulate *this matrix:
1336 it = this->m.begin();
1337 tmp_n_it = tmp_n.m.begin();
1338 tmp_d_it = tmp_d.m.begin();
1339 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
1340 (*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
1346 /** Partial pivoting method for matrix elimination schemes.
1347 * Usual pivoting (symbolic==false) returns the index to the element with the
1348 * largest absolute value in column ro and swaps the current row with the one
1349 * where the element was found. With (symbolic==true) it does the same thing
1350 * with the first non-zero element.
1352 * @param ro is the row from where to begin
1353 * @param co is the column to be inspected
1354 * @param symbolic signal if we want the first non-zero element to be pivoted
1355 * (true) or the one with the largest absolute value (false).
1356 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1357 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1359 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1363 // search first non-zero element in column co beginning at row ro
1364 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1367 // search largest element in column co beginning at row ro
1368 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1369 unsigned kmax = k+1;
1370 numeric mmax = abs(ex_to_numeric(m[kmax*col+co]));
1372 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1373 numeric tmp = ex_to_numeric(this->m[kmax*col+co]);
1374 if (abs(tmp) > mmax) {
1380 if (!mmax.is_zero())
1384 // all elements in column co below row ro vanish
1387 // matrix needs no pivoting
1389 // matrix needs pivoting, so swap rows k and ro
1390 ensure_if_modifiable();
1391 for (unsigned c=0; c<col; ++c)
1392 this->m[k*col+c].swap(this->m[ro*col+c]);
1397 ex lst_to_matrix(const lst & l)
1399 // Find number of rows and columns
1400 unsigned rows = l.nops(), cols = 0, i, j;
1401 for (i=0; i<rows; i++)
1402 if (l.op(i).nops() > cols)
1403 cols = l.op(i).nops();
1405 // Allocate and fill matrix
1406 matrix &m = *new matrix(rows, cols);
1407 m.setflag(status_flags::dynallocated);
1408 for (i=0; i<rows; i++)
1409 for (j=0; j<cols; j++)
1410 if (l.op(i).nops() > j)
1411 m.set(i, j, l.op(i).op(j));
1417 ex diag_matrix(const lst & l)
1419 unsigned dim = l.nops();
1421 matrix &m = *new matrix(dim, dim);
1422 m.setflag(status_flags::dynallocated);
1423 for (unsigned i=0; i<dim; i++)
1424 m.set(i, i, l.op(i));
1429 } // namespace GiNaC